The most successful treatment for aplastic anemia is bone marrow transplantation. However, few patients are eligible for this procedure. For others, treatment usually consists of immunosuppressive agents, such as antithymocyte globulin (ATG) and cyclosporine. Unfortunately, even with immunosuppressive therapy, relapse is common. New combinations of medications may offer alternative and more effective treatment options. Sirolimus and cyclosporine are two drugs routinely used to suppress the immune system and prevent rejection in patients who have received organ transplants. While cyclosporine has been proven effective for treating aplastic anemia, sirolimus has not been tested for this disease. This study will evaluate the safety and efficacy of sirolimus in combination with cyclosporine for treating individuals with aplastic anemia that has not responded to other treatments.